We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { minus_active(x, y) -> minus(x, y) , minus_active(0(), y) -> 0() , minus_active(s(x), s(y)) -> minus_active(x, y) , mark(0()) -> 0() , mark(s(x)) -> s(mark(x)) , mark(minus(x, y)) -> minus_active(x, y) , mark(ge(x, y)) -> ge_active(x, y) , mark(div(x, y)) -> div_active(mark(x), y) , mark(if(x, y, z)) -> if_active(mark(x), y, z) , ge_active(x, y) -> ge(x, y) , ge_active(x, 0()) -> true() , ge_active(0(), s(y)) -> false() , ge_active(s(x), s(y)) -> ge_active(x, y) , div_active(x, y) -> div(x, y) , div_active(0(), s(y)) -> 0() , div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0()) , if_active(x, y, z) -> if(x, y, z) , if_active(true(), x, y) -> mark(x) , if_active(false(), x, y) -> mark(y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus_active](x1, x2) = [4] [0] = [0] [mark](x1) = [0] [s](x1) = [1] x1 + [0] [ge_active](x1, x2) = [0] [true] = [0] [minus](x1, x2) = [0] [false] = [0] [ge](x1, x2) = [0] [div](x1, x2) = [0] [div_active](x1, x2) = [1] x1 + [0] [if](x1, x2, x3) = [1] x1 + [0] [if_active](x1, x2, x3) = [1] x1 + [0] The order satisfies the following ordering constraints: [minus_active(x, y)] = [4] > [0] = [minus(x, y)] [minus_active(0(), y)] = [4] > [0] = [0()] [minus_active(s(x), s(y))] = [4] >= [4] = [minus_active(x, y)] [mark(0())] = [0] >= [0] = [0()] [mark(s(x))] = [0] >= [0] = [s(mark(x))] [mark(minus(x, y))] = [0] ? [4] = [minus_active(x, y)] [mark(ge(x, y))] = [0] >= [0] = [ge_active(x, y)] [mark(div(x, y))] = [0] >= [0] = [div_active(mark(x), y)] [mark(if(x, y, z))] = [0] >= [0] = [if_active(mark(x), y, z)] [ge_active(x, y)] = [0] >= [0] = [ge(x, y)] [ge_active(x, 0())] = [0] >= [0] = [true()] [ge_active(0(), s(y))] = [0] >= [0] = [false()] [ge_active(s(x), s(y))] = [0] >= [0] = [ge_active(x, y)] [div_active(x, y)] = [1] x + [0] >= [0] = [div(x, y)] [div_active(0(), s(y))] = [0] >= [0] = [0()] [div_active(s(x), s(y))] = [1] x + [0] >= [0] = [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())] [if_active(x, y, z)] = [1] x + [0] >= [1] x + [0] = [if(x, y, z)] [if_active(true(), x, y)] = [0] >= [0] = [mark(x)] [if_active(false(), x, y)] = [0] >= [0] = [mark(y)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { minus_active(s(x), s(y)) -> minus_active(x, y) , mark(0()) -> 0() , mark(s(x)) -> s(mark(x)) , mark(minus(x, y)) -> minus_active(x, y) , mark(ge(x, y)) -> ge_active(x, y) , mark(div(x, y)) -> div_active(mark(x), y) , mark(if(x, y, z)) -> if_active(mark(x), y, z) , ge_active(x, y) -> ge(x, y) , ge_active(x, 0()) -> true() , ge_active(0(), s(y)) -> false() , ge_active(s(x), s(y)) -> ge_active(x, y) , div_active(x, y) -> div(x, y) , div_active(0(), s(y)) -> 0() , div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0()) , if_active(x, y, z) -> if(x, y, z) , if_active(true(), x, y) -> mark(x) , if_active(false(), x, y) -> mark(y) } Weak Trs: { minus_active(x, y) -> minus(x, y) , minus_active(0(), y) -> 0() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus_active](x1, x2) = [4] [0] = [1] [mark](x1) = [1] x1 + [0] [s](x1) = [1] x1 + [0] [ge_active](x1, x2) = [1] x1 + [0] [true] = [0] [minus](x1, x2) = [0] [false] = [0] [ge](x1, x2) = [1] x1 + [4] [div](x1, x2) = [1] x1 + [1] x2 + [0] [div_active](x1, x2) = [1] x1 + [1] x2 + [0] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] [if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The order satisfies the following ordering constraints: [minus_active(x, y)] = [4] > [0] = [minus(x, y)] [minus_active(0(), y)] = [4] > [1] = [0()] [minus_active(s(x), s(y))] = [4] >= [4] = [minus_active(x, y)] [mark(0())] = [1] >= [1] = [0()] [mark(s(x))] = [1] x + [0] >= [1] x + [0] = [s(mark(x))] [mark(minus(x, y))] = [0] ? [4] = [minus_active(x, y)] [mark(ge(x, y))] = [1] x + [4] > [1] x + [0] = [ge_active(x, y)] [mark(div(x, y))] = [1] y + [1] x + [0] >= [1] y + [1] x + [0] = [div_active(mark(x), y)] [mark(if(x, y, z))] = [1] y + [1] x + [1] z + [0] >= [1] y + [1] x + [1] z + [0] = [if_active(mark(x), y, z)] [ge_active(x, y)] = [1] x + [0] ? [1] x + [4] = [ge(x, y)] [ge_active(x, 0())] = [1] x + [0] >= [0] = [true()] [ge_active(0(), s(y))] = [1] > [0] = [false()] [ge_active(s(x), s(y))] = [1] x + [0] >= [1] x + [0] = [ge_active(x, y)] [div_active(x, y)] = [1] y + [1] x + [0] >= [1] y + [1] x + [0] = [div(x, y)] [div_active(0(), s(y))] = [1] y + [1] >= [1] = [0()] [div_active(s(x), s(y))] = [1] y + [1] x + [0] ? [1] y + [1] x + [1] = [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())] [if_active(x, y, z)] = [1] y + [1] x + [1] z + [0] >= [1] y + [1] x + [1] z + [0] = [if(x, y, z)] [if_active(true(), x, y)] = [1] y + [1] x + [0] >= [1] x + [0] = [mark(x)] [if_active(false(), x, y)] = [1] y + [1] x + [0] >= [1] y + [0] = [mark(y)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { minus_active(s(x), s(y)) -> minus_active(x, y) , mark(0()) -> 0() , mark(s(x)) -> s(mark(x)) , mark(minus(x, y)) -> minus_active(x, y) , mark(div(x, y)) -> div_active(mark(x), y) , mark(if(x, y, z)) -> if_active(mark(x), y, z) , ge_active(x, y) -> ge(x, y) , ge_active(x, 0()) -> true() , ge_active(s(x), s(y)) -> ge_active(x, y) , div_active(x, y) -> div(x, y) , div_active(0(), s(y)) -> 0() , div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0()) , if_active(x, y, z) -> if(x, y, z) , if_active(true(), x, y) -> mark(x) , if_active(false(), x, y) -> mark(y) } Weak Trs: { minus_active(x, y) -> minus(x, y) , minus_active(0(), y) -> 0() , mark(ge(x, y)) -> ge_active(x, y) , ge_active(0(), s(y)) -> false() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus_active](x1, x2) = [4] [0] = [0] [mark](x1) = [1] x1 + [1] [s](x1) = [1] x1 + [0] [ge_active](x1, x2) = [1] x1 + [5] [true] = [0] [minus](x1, x2) = [3] [false] = [0] [ge](x1, x2) = [1] x1 + [7] [div](x1, x2) = [1] x1 + [0] [div_active](x1, x2) = [1] x1 + [0] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] [if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The order satisfies the following ordering constraints: [minus_active(x, y)] = [4] > [3] = [minus(x, y)] [minus_active(0(), y)] = [4] > [0] = [0()] [minus_active(s(x), s(y))] = [4] >= [4] = [minus_active(x, y)] [mark(0())] = [1] > [0] = [0()] [mark(s(x))] = [1] x + [1] >= [1] x + [1] = [s(mark(x))] [mark(minus(x, y))] = [4] >= [4] = [minus_active(x, y)] [mark(ge(x, y))] = [1] x + [8] > [1] x + [5] = [ge_active(x, y)] [mark(div(x, y))] = [1] x + [1] >= [1] x + [1] = [div_active(mark(x), y)] [mark(if(x, y, z))] = [1] y + [1] x + [1] z + [1] >= [1] y + [1] x + [1] z + [1] = [if_active(mark(x), y, z)] [ge_active(x, y)] = [1] x + [5] ? [1] x + [7] = [ge(x, y)] [ge_active(x, 0())] = [1] x + [5] > [0] = [true()] [ge_active(0(), s(y))] = [5] > [0] = [false()] [ge_active(s(x), s(y))] = [1] x + [5] >= [1] x + [5] = [ge_active(x, y)] [div_active(x, y)] = [1] x + [0] >= [1] x + [0] = [div(x, y)] [div_active(0(), s(y))] = [0] >= [0] = [0()] [div_active(s(x), s(y))] = [1] x + [0] ? [1] x + [8] = [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())] [if_active(x, y, z)] = [1] y + [1] x + [1] z + [0] >= [1] y + [1] x + [1] z + [0] = [if(x, y, z)] [if_active(true(), x, y)] = [1] y + [1] x + [0] ? [1] x + [1] = [mark(x)] [if_active(false(), x, y)] = [1] y + [1] x + [0] ? [1] y + [1] = [mark(y)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { minus_active(s(x), s(y)) -> minus_active(x, y) , mark(s(x)) -> s(mark(x)) , mark(minus(x, y)) -> minus_active(x, y) , mark(div(x, y)) -> div_active(mark(x), y) , mark(if(x, y, z)) -> if_active(mark(x), y, z) , ge_active(x, y) -> ge(x, y) , ge_active(s(x), s(y)) -> ge_active(x, y) , div_active(x, y) -> div(x, y) , div_active(0(), s(y)) -> 0() , div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0()) , if_active(x, y, z) -> if(x, y, z) , if_active(true(), x, y) -> mark(x) , if_active(false(), x, y) -> mark(y) } Weak Trs: { minus_active(x, y) -> minus(x, y) , minus_active(0(), y) -> 0() , mark(0()) -> 0() , mark(ge(x, y)) -> ge_active(x, y) , ge_active(x, 0()) -> true() , ge_active(0(), s(y)) -> false() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus_active](x1, x2) = [4] [0] = [1] [mark](x1) = [1] x1 + [1] [s](x1) = [1] x1 + [0] [ge_active](x1, x2) = [0] [true] = [0] [minus](x1, x2) = [0] [false] = [0] [ge](x1, x2) = [0] [div](x1, x2) = [1] x1 + [0] [div_active](x1, x2) = [1] x1 + [0] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [7] [if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The order satisfies the following ordering constraints: [minus_active(x, y)] = [4] > [0] = [minus(x, y)] [minus_active(0(), y)] = [4] > [1] = [0()] [minus_active(s(x), s(y))] = [4] >= [4] = [minus_active(x, y)] [mark(0())] = [2] > [1] = [0()] [mark(s(x))] = [1] x + [1] >= [1] x + [1] = [s(mark(x))] [mark(minus(x, y))] = [1] ? [4] = [minus_active(x, y)] [mark(ge(x, y))] = [1] > [0] = [ge_active(x, y)] [mark(div(x, y))] = [1] x + [1] >= [1] x + [1] = [div_active(mark(x), y)] [mark(if(x, y, z))] = [1] y + [1] x + [1] z + [8] > [1] y + [1] x + [1] z + [1] = [if_active(mark(x), y, z)] [ge_active(x, y)] = [0] >= [0] = [ge(x, y)] [ge_active(x, 0())] = [0] >= [0] = [true()] [ge_active(0(), s(y))] = [0] >= [0] = [false()] [ge_active(s(x), s(y))] = [0] >= [0] = [ge_active(x, y)] [div_active(x, y)] = [1] x + [0] >= [1] x + [0] = [div(x, y)] [div_active(0(), s(y))] = [1] >= [1] = [0()] [div_active(s(x), s(y))] = [1] x + [0] ? [1] = [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())] [if_active(x, y, z)] = [1] y + [1] x + [1] z + [0] ? [1] y + [1] x + [1] z + [7] = [if(x, y, z)] [if_active(true(), x, y)] = [1] y + [1] x + [0] ? [1] x + [1] = [mark(x)] [if_active(false(), x, y)] = [1] y + [1] x + [0] ? [1] y + [1] = [mark(y)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { minus_active(s(x), s(y)) -> minus_active(x, y) , mark(s(x)) -> s(mark(x)) , mark(minus(x, y)) -> minus_active(x, y) , mark(div(x, y)) -> div_active(mark(x), y) , ge_active(x, y) -> ge(x, y) , ge_active(s(x), s(y)) -> ge_active(x, y) , div_active(x, y) -> div(x, y) , div_active(0(), s(y)) -> 0() , div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0()) , if_active(x, y, z) -> if(x, y, z) , if_active(true(), x, y) -> mark(x) , if_active(false(), x, y) -> mark(y) } Weak Trs: { minus_active(x, y) -> minus(x, y) , minus_active(0(), y) -> 0() , mark(0()) -> 0() , mark(ge(x, y)) -> ge_active(x, y) , mark(if(x, y, z)) -> if_active(mark(x), y, z) , ge_active(x, 0()) -> true() , ge_active(0(), s(y)) -> false() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus_active](x1, x2) = [4] [0] = [0] [mark](x1) = [1] x1 + [1] [s](x1) = [1] x1 + [0] [ge_active](x1, x2) = [5] [true] = [0] [minus](x1, x2) = [0] [false] = [0] [ge](x1, x2) = [7] [div](x1, x2) = [1] x1 + [7] [div_active](x1, x2) = [1] x1 + [0] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [7] [if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The order satisfies the following ordering constraints: [minus_active(x, y)] = [4] > [0] = [minus(x, y)] [minus_active(0(), y)] = [4] > [0] = [0()] [minus_active(s(x), s(y))] = [4] >= [4] = [minus_active(x, y)] [mark(0())] = [1] > [0] = [0()] [mark(s(x))] = [1] x + [1] >= [1] x + [1] = [s(mark(x))] [mark(minus(x, y))] = [1] ? [4] = [minus_active(x, y)] [mark(ge(x, y))] = [8] > [5] = [ge_active(x, y)] [mark(div(x, y))] = [1] x + [8] > [1] x + [1] = [div_active(mark(x), y)] [mark(if(x, y, z))] = [1] y + [1] x + [1] z + [8] > [1] y + [1] x + [1] z + [1] = [if_active(mark(x), y, z)] [ge_active(x, y)] = [5] ? [7] = [ge(x, y)] [ge_active(x, 0())] = [5] > [0] = [true()] [ge_active(0(), s(y))] = [5] > [0] = [false()] [ge_active(s(x), s(y))] = [5] >= [5] = [ge_active(x, y)] [div_active(x, y)] = [1] x + [0] ? [1] x + [7] = [div(x, y)] [div_active(0(), s(y))] = [0] >= [0] = [0()] [div_active(s(x), s(y))] = [1] x + [0] ? [12] = [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())] [if_active(x, y, z)] = [1] y + [1] x + [1] z + [0] ? [1] y + [1] x + [1] z + [7] = [if(x, y, z)] [if_active(true(), x, y)] = [1] y + [1] x + [0] ? [1] x + [1] = [mark(x)] [if_active(false(), x, y)] = [1] y + [1] x + [0] ? [1] y + [1] = [mark(y)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { minus_active(s(x), s(y)) -> minus_active(x, y) , mark(s(x)) -> s(mark(x)) , mark(minus(x, y)) -> minus_active(x, y) , ge_active(x, y) -> ge(x, y) , ge_active(s(x), s(y)) -> ge_active(x, y) , div_active(x, y) -> div(x, y) , div_active(0(), s(y)) -> 0() , div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0()) , if_active(x, y, z) -> if(x, y, z) , if_active(true(), x, y) -> mark(x) , if_active(false(), x, y) -> mark(y) } Weak Trs: { minus_active(x, y) -> minus(x, y) , minus_active(0(), y) -> 0() , mark(0()) -> 0() , mark(ge(x, y)) -> ge_active(x, y) , mark(div(x, y)) -> div_active(mark(x), y) , mark(if(x, y, z)) -> if_active(mark(x), y, z) , ge_active(x, 0()) -> true() , ge_active(0(), s(y)) -> false() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus_active](x1, x2) = [0] [0] = [0] [mark](x1) = [1] x1 + [1] [s](x1) = [1] x1 + [0] [ge_active](x1, x2) = [5] [true] = [0] [minus](x1, x2) = [0] [false] = [0] [ge](x1, x2) = [7] [div](x1, x2) = [1] x1 + [7] [div_active](x1, x2) = [1] x1 + [0] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [7] [if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The order satisfies the following ordering constraints: [minus_active(x, y)] = [0] >= [0] = [minus(x, y)] [minus_active(0(), y)] = [0] >= [0] = [0()] [minus_active(s(x), s(y))] = [0] >= [0] = [minus_active(x, y)] [mark(0())] = [1] > [0] = [0()] [mark(s(x))] = [1] x + [1] >= [1] x + [1] = [s(mark(x))] [mark(minus(x, y))] = [1] > [0] = [minus_active(x, y)] [mark(ge(x, y))] = [8] > [5] = [ge_active(x, y)] [mark(div(x, y))] = [1] x + [8] > [1] x + [1] = [div_active(mark(x), y)] [mark(if(x, y, z))] = [1] y + [1] x + [1] z + [8] > [1] y + [1] x + [1] z + [1] = [if_active(mark(x), y, z)] [ge_active(x, y)] = [5] ? [7] = [ge(x, y)] [ge_active(x, 0())] = [5] > [0] = [true()] [ge_active(0(), s(y))] = [5] > [0] = [false()] [ge_active(s(x), s(y))] = [5] >= [5] = [ge_active(x, y)] [div_active(x, y)] = [1] x + [0] ? [1] x + [7] = [div(x, y)] [div_active(0(), s(y))] = [0] >= [0] = [0()] [div_active(s(x), s(y))] = [1] x + [0] ? [12] = [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())] [if_active(x, y, z)] = [1] y + [1] x + [1] z + [0] ? [1] y + [1] x + [1] z + [7] = [if(x, y, z)] [if_active(true(), x, y)] = [1] y + [1] x + [0] ? [1] x + [1] = [mark(x)] [if_active(false(), x, y)] = [1] y + [1] x + [0] ? [1] y + [1] = [mark(y)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { minus_active(s(x), s(y)) -> minus_active(x, y) , mark(s(x)) -> s(mark(x)) , ge_active(x, y) -> ge(x, y) , ge_active(s(x), s(y)) -> ge_active(x, y) , div_active(x, y) -> div(x, y) , div_active(0(), s(y)) -> 0() , div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0()) , if_active(x, y, z) -> if(x, y, z) , if_active(true(), x, y) -> mark(x) , if_active(false(), x, y) -> mark(y) } Weak Trs: { minus_active(x, y) -> minus(x, y) , minus_active(0(), y) -> 0() , mark(0()) -> 0() , mark(minus(x, y)) -> minus_active(x, y) , mark(ge(x, y)) -> ge_active(x, y) , mark(div(x, y)) -> div_active(mark(x), y) , mark(if(x, y, z)) -> if_active(mark(x), y, z) , ge_active(x, 0()) -> true() , ge_active(0(), s(y)) -> false() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus_active](x1, x2) = [1] x1 + [0] [0] = [4] [mark](x1) = [1] x1 + [0] [s](x1) = [1] x1 + [0] [ge_active](x1, x2) = [4] [true] = [3] [minus](x1, x2) = [1] x1 + [0] [false] = [0] [ge](x1, x2) = [4] [div](x1, x2) = [1] x1 + [1] x2 + [0] [div_active](x1, x2) = [1] x1 + [1] x2 + [0] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [4] [if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1] The order satisfies the following ordering constraints: [minus_active(x, y)] = [1] x + [0] >= [1] x + [0] = [minus(x, y)] [minus_active(0(), y)] = [4] >= [4] = [0()] [minus_active(s(x), s(y))] = [1] x + [0] >= [1] x + [0] = [minus_active(x, y)] [mark(0())] = [4] >= [4] = [0()] [mark(s(x))] = [1] x + [0] >= [1] x + [0] = [s(mark(x))] [mark(minus(x, y))] = [1] x + [0] >= [1] x + [0] = [minus_active(x, y)] [mark(ge(x, y))] = [4] >= [4] = [ge_active(x, y)] [mark(div(x, y))] = [1] y + [1] x + [0] >= [1] y + [1] x + [0] = [div_active(mark(x), y)] [mark(if(x, y, z))] = [1] y + [1] x + [1] z + [4] > [1] y + [1] x + [1] z + [1] = [if_active(mark(x), y, z)] [ge_active(x, y)] = [4] >= [4] = [ge(x, y)] [ge_active(x, 0())] = [4] > [3] = [true()] [ge_active(0(), s(y))] = [4] > [0] = [false()] [ge_active(s(x), s(y))] = [4] >= [4] = [ge_active(x, y)] [div_active(x, y)] = [1] y + [1] x + [0] >= [1] y + [1] x + [0] = [div(x, y)] [div_active(0(), s(y))] = [1] y + [4] >= [4] = [0()] [div_active(s(x), s(y))] = [1] y + [1] x + [0] ? [1] y + [1] x + [9] = [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())] [if_active(x, y, z)] = [1] y + [1] x + [1] z + [1] ? [1] y + [1] x + [1] z + [4] = [if(x, y, z)] [if_active(true(), x, y)] = [1] y + [1] x + [4] > [1] x + [0] = [mark(x)] [if_active(false(), x, y)] = [1] y + [1] x + [1] > [1] y + [0] = [mark(y)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { minus_active(s(x), s(y)) -> minus_active(x, y) , mark(s(x)) -> s(mark(x)) , ge_active(x, y) -> ge(x, y) , ge_active(s(x), s(y)) -> ge_active(x, y) , div_active(x, y) -> div(x, y) , div_active(0(), s(y)) -> 0() , div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0()) , if_active(x, y, z) -> if(x, y, z) } Weak Trs: { minus_active(x, y) -> minus(x, y) , minus_active(0(), y) -> 0() , mark(0()) -> 0() , mark(minus(x, y)) -> minus_active(x, y) , mark(ge(x, y)) -> ge_active(x, y) , mark(div(x, y)) -> div_active(mark(x), y) , mark(if(x, y, z)) -> if_active(mark(x), y, z) , ge_active(x, 0()) -> true() , ge_active(0(), s(y)) -> false() , if_active(true(), x, y) -> mark(x) , if_active(false(), x, y) -> mark(y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus_active](x1, x2) = [4] [0] = [0] [mark](x1) = [1] x1 + [0] [s](x1) = [1] x1 + [0] [ge_active](x1, x2) = [1] x1 + [4] [true] = [0] [minus](x1, x2) = [4] [false] = [4] [ge](x1, x2) = [1] x1 + [5] [div](x1, x2) = [1] x1 + [1] [div_active](x1, x2) = [1] x1 + [1] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] [if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The order satisfies the following ordering constraints: [minus_active(x, y)] = [4] >= [4] = [minus(x, y)] [minus_active(0(), y)] = [4] > [0] = [0()] [minus_active(s(x), s(y))] = [4] >= [4] = [minus_active(x, y)] [mark(0())] = [0] >= [0] = [0()] [mark(s(x))] = [1] x + [0] >= [1] x + [0] = [s(mark(x))] [mark(minus(x, y))] = [4] >= [4] = [minus_active(x, y)] [mark(ge(x, y))] = [1] x + [5] > [1] x + [4] = [ge_active(x, y)] [mark(div(x, y))] = [1] x + [1] >= [1] x + [1] = [div_active(mark(x), y)] [mark(if(x, y, z))] = [1] y + [1] x + [1] z + [0] >= [1] y + [1] x + [1] z + [0] = [if_active(mark(x), y, z)] [ge_active(x, y)] = [1] x + [4] ? [1] x + [5] = [ge(x, y)] [ge_active(x, 0())] = [1] x + [4] > [0] = [true()] [ge_active(0(), s(y))] = [4] >= [4] = [false()] [ge_active(s(x), s(y))] = [1] x + [4] >= [1] x + [4] = [ge_active(x, y)] [div_active(x, y)] = [1] x + [1] >= [1] x + [1] = [div(x, y)] [div_active(0(), s(y))] = [1] > [0] = [0()] [div_active(s(x), s(y))] = [1] x + [1] ? [1] x + [9] = [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())] [if_active(x, y, z)] = [1] y + [1] x + [1] z + [0] >= [1] y + [1] x + [1] z + [0] = [if(x, y, z)] [if_active(true(), x, y)] = [1] y + [1] x + [0] >= [1] x + [0] = [mark(x)] [if_active(false(), x, y)] = [1] y + [1] x + [4] > [1] y + [0] = [mark(y)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { minus_active(s(x), s(y)) -> minus_active(x, y) , mark(s(x)) -> s(mark(x)) , ge_active(x, y) -> ge(x, y) , ge_active(s(x), s(y)) -> ge_active(x, y) , div_active(x, y) -> div(x, y) , div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0()) , if_active(x, y, z) -> if(x, y, z) } Weak Trs: { minus_active(x, y) -> minus(x, y) , minus_active(0(), y) -> 0() , mark(0()) -> 0() , mark(minus(x, y)) -> minus_active(x, y) , mark(ge(x, y)) -> ge_active(x, y) , mark(div(x, y)) -> div_active(mark(x), y) , mark(if(x, y, z)) -> if_active(mark(x), y, z) , ge_active(x, 0()) -> true() , ge_active(0(), s(y)) -> false() , div_active(0(), s(y)) -> 0() , if_active(true(), x, y) -> mark(x) , if_active(false(), x, y) -> mark(y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus_active](x1, x2) = [4] [0] = [4] [mark](x1) = [1] x1 + [1] [s](x1) = [1] x1 + [0] [ge_active](x1, x2) = [1] x1 + [1] [true] = [1] [minus](x1, x2) = [3] [false] = [4] [ge](x1, x2) = [1] x1 + [0] [div](x1, x2) = [1] x1 + [0] [div_active](x1, x2) = [1] x1 + [0] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] [if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The order satisfies the following ordering constraints: [minus_active(x, y)] = [4] > [3] = [minus(x, y)] [minus_active(0(), y)] = [4] >= [4] = [0()] [minus_active(s(x), s(y))] = [4] >= [4] = [minus_active(x, y)] [mark(0())] = [5] > [4] = [0()] [mark(s(x))] = [1] x + [1] >= [1] x + [1] = [s(mark(x))] [mark(minus(x, y))] = [4] >= [4] = [minus_active(x, y)] [mark(ge(x, y))] = [1] x + [1] >= [1] x + [1] = [ge_active(x, y)] [mark(div(x, y))] = [1] x + [1] >= [1] x + [1] = [div_active(mark(x), y)] [mark(if(x, y, z))] = [1] y + [1] x + [1] z + [1] >= [1] y + [1] x + [1] z + [1] = [if_active(mark(x), y, z)] [ge_active(x, y)] = [1] x + [1] > [1] x + [0] = [ge(x, y)] [ge_active(x, 0())] = [1] x + [1] >= [1] = [true()] [ge_active(0(), s(y))] = [5] > [4] = [false()] [ge_active(s(x), s(y))] = [1] x + [1] >= [1] x + [1] = [ge_active(x, y)] [div_active(x, y)] = [1] x + [0] >= [1] x + [0] = [div(x, y)] [div_active(0(), s(y))] = [4] >= [4] = [0()] [div_active(s(x), s(y))] = [1] x + [0] ? [1] x + [8] = [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())] [if_active(x, y, z)] = [1] y + [1] x + [1] z + [0] >= [1] y + [1] x + [1] z + [0] = [if(x, y, z)] [if_active(true(), x, y)] = [1] y + [1] x + [1] >= [1] x + [1] = [mark(x)] [if_active(false(), x, y)] = [1] y + [1] x + [4] > [1] y + [1] = [mark(y)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { minus_active(s(x), s(y)) -> minus_active(x, y) , mark(s(x)) -> s(mark(x)) , ge_active(s(x), s(y)) -> ge_active(x, y) , div_active(x, y) -> div(x, y) , div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0()) , if_active(x, y, z) -> if(x, y, z) } Weak Trs: { minus_active(x, y) -> minus(x, y) , minus_active(0(), y) -> 0() , mark(0()) -> 0() , mark(minus(x, y)) -> minus_active(x, y) , mark(ge(x, y)) -> ge_active(x, y) , mark(div(x, y)) -> div_active(mark(x), y) , mark(if(x, y, z)) -> if_active(mark(x), y, z) , ge_active(x, y) -> ge(x, y) , ge_active(x, 0()) -> true() , ge_active(0(), s(y)) -> false() , div_active(0(), s(y)) -> 0() , if_active(true(), x, y) -> mark(x) , if_active(false(), x, y) -> mark(y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus_active](x1, x2) = [0] [0] = [0] [mark](x1) = [0] [s](x1) = [1] x1 + [7] [ge_active](x1, x2) = [0] [true] = [0] [minus](x1, x2) = [0] [false] = [0] [ge](x1, x2) = [0] [div](x1, x2) = [1] x1 + [0] [div_active](x1, x2) = [1] x1 + [0] [if](x1, x2, x3) = [1] x1 + [0] [if_active](x1, x2, x3) = [1] x1 + [0] The order satisfies the following ordering constraints: [minus_active(x, y)] = [0] >= [0] = [minus(x, y)] [minus_active(0(), y)] = [0] >= [0] = [0()] [minus_active(s(x), s(y))] = [0] >= [0] = [minus_active(x, y)] [mark(0())] = [0] >= [0] = [0()] [mark(s(x))] = [0] ? [7] = [s(mark(x))] [mark(minus(x, y))] = [0] >= [0] = [minus_active(x, y)] [mark(ge(x, y))] = [0] >= [0] = [ge_active(x, y)] [mark(div(x, y))] = [0] >= [0] = [div_active(mark(x), y)] [mark(if(x, y, z))] = [0] >= [0] = [if_active(mark(x), y, z)] [ge_active(x, y)] = [0] >= [0] = [ge(x, y)] [ge_active(x, 0())] = [0] >= [0] = [true()] [ge_active(0(), s(y))] = [0] >= [0] = [false()] [ge_active(s(x), s(y))] = [0] >= [0] = [ge_active(x, y)] [div_active(x, y)] = [1] x + [0] >= [1] x + [0] = [div(x, y)] [div_active(0(), s(y))] = [0] >= [0] = [0()] [div_active(s(x), s(y))] = [1] x + [7] > [0] = [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())] [if_active(x, y, z)] = [1] x + [0] >= [1] x + [0] = [if(x, y, z)] [if_active(true(), x, y)] = [0] >= [0] = [mark(x)] [if_active(false(), x, y)] = [0] >= [0] = [mark(y)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { minus_active(s(x), s(y)) -> minus_active(x, y) , mark(s(x)) -> s(mark(x)) , ge_active(s(x), s(y)) -> ge_active(x, y) , div_active(x, y) -> div(x, y) , if_active(x, y, z) -> if(x, y, z) } Weak Trs: { minus_active(x, y) -> minus(x, y) , minus_active(0(), y) -> 0() , mark(0()) -> 0() , mark(minus(x, y)) -> minus_active(x, y) , mark(ge(x, y)) -> ge_active(x, y) , mark(div(x, y)) -> div_active(mark(x), y) , mark(if(x, y, z)) -> if_active(mark(x), y, z) , ge_active(x, y) -> ge(x, y) , ge_active(x, 0()) -> true() , ge_active(0(), s(y)) -> false() , div_active(0(), s(y)) -> 0() , div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0()) , if_active(true(), x, y) -> mark(x) , if_active(false(), x, y) -> mark(y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus_active](x1, x2) = [0] [0] = [0] [mark](x1) = [1] x1 + [0] [s](x1) = [1] x1 + [1] [ge_active](x1, x2) = [1] x1 + [0] [true] = [0] [minus](x1, x2) = [0] [false] = [0] [ge](x1, x2) = [1] x1 + [0] [div](x1, x2) = [1] x1 + [0] [div_active](x1, x2) = [1] x1 + [0] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] [if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The order satisfies the following ordering constraints: [minus_active(x, y)] = [0] >= [0] = [minus(x, y)] [minus_active(0(), y)] = [0] >= [0] = [0()] [minus_active(s(x), s(y))] = [0] >= [0] = [minus_active(x, y)] [mark(0())] = [0] >= [0] = [0()] [mark(s(x))] = [1] x + [1] >= [1] x + [1] = [s(mark(x))] [mark(minus(x, y))] = [0] >= [0] = [minus_active(x, y)] [mark(ge(x, y))] = [1] x + [0] >= [1] x + [0] = [ge_active(x, y)] [mark(div(x, y))] = [1] x + [0] >= [1] x + [0] = [div_active(mark(x), y)] [mark(if(x, y, z))] = [1] y + [1] x + [1] z + [0] >= [1] y + [1] x + [1] z + [0] = [if_active(mark(x), y, z)] [ge_active(x, y)] = [1] x + [0] >= [1] x + [0] = [ge(x, y)] [ge_active(x, 0())] = [1] x + [0] >= [0] = [true()] [ge_active(0(), s(y))] = [0] >= [0] = [false()] [ge_active(s(x), s(y))] = [1] x + [1] > [1] x + [0] = [ge_active(x, y)] [div_active(x, y)] = [1] x + [0] >= [1] x + [0] = [div(x, y)] [div_active(0(), s(y))] = [0] >= [0] = [0()] [div_active(s(x), s(y))] = [1] x + [1] >= [1] x + [1] = [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())] [if_active(x, y, z)] = [1] y + [1] x + [1] z + [0] >= [1] y + [1] x + [1] z + [0] = [if(x, y, z)] [if_active(true(), x, y)] = [1] y + [1] x + [0] >= [1] x + [0] = [mark(x)] [if_active(false(), x, y)] = [1] y + [1] x + [0] >= [1] y + [0] = [mark(y)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { minus_active(s(x), s(y)) -> minus_active(x, y) , mark(s(x)) -> s(mark(x)) , div_active(x, y) -> div(x, y) , if_active(x, y, z) -> if(x, y, z) } Weak Trs: { minus_active(x, y) -> minus(x, y) , minus_active(0(), y) -> 0() , mark(0()) -> 0() , mark(minus(x, y)) -> minus_active(x, y) , mark(ge(x, y)) -> ge_active(x, y) , mark(div(x, y)) -> div_active(mark(x), y) , mark(if(x, y, z)) -> if_active(mark(x), y, z) , ge_active(x, y) -> ge(x, y) , ge_active(x, 0()) -> true() , ge_active(0(), s(y)) -> false() , ge_active(s(x), s(y)) -> ge_active(x, y) , div_active(0(), s(y)) -> 0() , div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0()) , if_active(true(), x, y) -> mark(x) , if_active(false(), x, y) -> mark(y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus_active](x1, x2) = [1] x1 + [0] [0] = [0] [mark](x1) = [1] x1 + [0] [s](x1) = [1] x1 + [3] [ge_active](x1, x2) = [0] [true] = [0] [minus](x1, x2) = [1] x1 + [0] [false] = [0] [ge](x1, x2) = [0] [div](x1, x2) = [1] x1 + [1] x2 + [6] [div_active](x1, x2) = [1] x1 + [1] x2 + [6] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [4] [if_active](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The order satisfies the following ordering constraints: [minus_active(x, y)] = [1] x + [0] >= [1] x + [0] = [minus(x, y)] [minus_active(0(), y)] = [0] >= [0] = [0()] [minus_active(s(x), s(y))] = [1] x + [3] > [1] x + [0] = [minus_active(x, y)] [mark(0())] = [0] >= [0] = [0()] [mark(s(x))] = [1] x + [3] >= [1] x + [3] = [s(mark(x))] [mark(minus(x, y))] = [1] x + [0] >= [1] x + [0] = [minus_active(x, y)] [mark(ge(x, y))] = [0] >= [0] = [ge_active(x, y)] [mark(div(x, y))] = [1] y + [1] x + [6] >= [1] y + [1] x + [6] = [div_active(mark(x), y)] [mark(if(x, y, z))] = [1] y + [1] x + [1] z + [4] > [1] y + [1] x + [1] z + [0] = [if_active(mark(x), y, z)] [ge_active(x, y)] = [0] >= [0] = [ge(x, y)] [ge_active(x, 0())] = [0] >= [0] = [true()] [ge_active(0(), s(y))] = [0] >= [0] = [false()] [ge_active(s(x), s(y))] = [0] >= [0] = [ge_active(x, y)] [div_active(x, y)] = [1] y + [1] x + [6] >= [1] y + [1] x + [6] = [div(x, y)] [div_active(0(), s(y))] = [1] y + [9] > [0] = [0()] [div_active(s(x), s(y))] = [1] y + [1] x + [12] >= [1] y + [1] x + [12] = [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())] [if_active(x, y, z)] = [1] y + [1] x + [1] z + [0] ? [1] y + [1] x + [1] z + [4] = [if(x, y, z)] [if_active(true(), x, y)] = [1] y + [1] x + [0] >= [1] x + [0] = [mark(x)] [if_active(false(), x, y)] = [1] y + [1] x + [0] >= [1] y + [0] = [mark(y)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { mark(s(x)) -> s(mark(x)) , div_active(x, y) -> div(x, y) , if_active(x, y, z) -> if(x, y, z) } Weak Trs: { minus_active(x, y) -> minus(x, y) , minus_active(0(), y) -> 0() , minus_active(s(x), s(y)) -> minus_active(x, y) , mark(0()) -> 0() , mark(minus(x, y)) -> minus_active(x, y) , mark(ge(x, y)) -> ge_active(x, y) , mark(div(x, y)) -> div_active(mark(x), y) , mark(if(x, y, z)) -> if_active(mark(x), y, z) , ge_active(x, y) -> ge(x, y) , ge_active(x, 0()) -> true() , ge_active(0(), s(y)) -> false() , ge_active(s(x), s(y)) -> ge_active(x, y) , div_active(0(), s(y)) -> 0() , div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0()) , if_active(true(), x, y) -> mark(x) , if_active(false(), x, y) -> mark(y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. Trs: { div_active(x, y) -> div(x, y) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [minus_active](x1, x2) = [0] [0] = [0] [mark](x1) = [5] x1 + [0] [s](x1) = [1] x1 + [0] [ge_active](x1, x2) = [0] [true] = [0] [minus](x1, x2) = [0] [false] = [0] [ge](x1, x2) = [0] [div](x1, x2) = [1] x1 + [1] x2 + [1] [div_active](x1, x2) = [1] x1 + [5] x2 + [5] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] [if_active](x1, x2, x3) = [1] x1 + [5] x2 + [5] x3 + [0] The order satisfies the following ordering constraints: [minus_active(x, y)] = [0] >= [0] = [minus(x, y)] [minus_active(0(), y)] = [0] >= [0] = [0()] [minus_active(s(x), s(y))] = [0] >= [0] = [minus_active(x, y)] [mark(0())] = [0] >= [0] = [0()] [mark(s(x))] = [5] x + [0] >= [5] x + [0] = [s(mark(x))] [mark(minus(x, y))] = [0] >= [0] = [minus_active(x, y)] [mark(ge(x, y))] = [0] >= [0] = [ge_active(x, y)] [mark(div(x, y))] = [5] y + [5] x + [5] >= [5] y + [5] x + [5] = [div_active(mark(x), y)] [mark(if(x, y, z))] = [5] y + [5] x + [5] z + [0] >= [5] y + [5] x + [5] z + [0] = [if_active(mark(x), y, z)] [ge_active(x, y)] = [0] >= [0] = [ge(x, y)] [ge_active(x, 0())] = [0] >= [0] = [true()] [ge_active(0(), s(y))] = [0] >= [0] = [false()] [ge_active(s(x), s(y))] = [0] >= [0] = [ge_active(x, y)] [div_active(x, y)] = [5] y + [1] x + [5] > [1] y + [1] x + [1] = [div(x, y)] [div_active(0(), s(y))] = [5] y + [5] > [0] = [0()] [div_active(s(x), s(y))] = [5] y + [1] x + [5] >= [5] y + [5] = [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())] [if_active(x, y, z)] = [5] y + [1] x + [5] z + [0] >= [1] y + [1] x + [1] z + [0] = [if(x, y, z)] [if_active(true(), x, y)] = [5] y + [5] x + [0] >= [5] x + [0] = [mark(x)] [if_active(false(), x, y)] = [5] y + [5] x + [0] >= [5] y + [0] = [mark(y)] We return to the main proof. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { mark(s(x)) -> s(mark(x)) , if_active(x, y, z) -> if(x, y, z) } Weak Trs: { minus_active(x, y) -> minus(x, y) , minus_active(0(), y) -> 0() , minus_active(s(x), s(y)) -> minus_active(x, y) , mark(0()) -> 0() , mark(minus(x, y)) -> minus_active(x, y) , mark(ge(x, y)) -> ge_active(x, y) , mark(div(x, y)) -> div_active(mark(x), y) , mark(if(x, y, z)) -> if_active(mark(x), y, z) , ge_active(x, y) -> ge(x, y) , ge_active(x, 0()) -> true() , ge_active(0(), s(y)) -> false() , ge_active(s(x), s(y)) -> ge_active(x, y) , div_active(x, y) -> div(x, y) , div_active(0(), s(y)) -> 0() , div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0()) , if_active(true(), x, y) -> mark(x) , if_active(false(), x, y) -> mark(y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 2' to orient following rules strictly. Trs: { mark(s(x)) -> s(mark(x)) , if_active(x, y, z) -> if(x, y, z) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(s) = {1}, Uargs(div_active) = {1}, Uargs(if_active) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus_active](x1, x2) = [0] [0] [0] = [0] [0] [mark](x1) = [2 0] x1 + [0] [0 2] [0] [s](x1) = [1 0] x1 + [3] [0 0] [4] [ge_active](x1, x2) = [1] [0] [true] = [1] [0] [minus](x1, x2) = [0] [0] [false] = [0] [0] [ge](x1, x2) = [1] [0] [div](x1, x2) = [1 3] x1 + [0 0] x2 + [0] [0 0] [0 1] [0] [div_active](x1, x2) = [1 3] x1 + [0 0] x2 + [0] [0 0] [0 2] [0] [if](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [3] [0 1] [0 1] [0 1] [0] [if_active](x1, x2, x3) = [1 0] x1 + [2 0] x2 + [2 0] x3 + [5] [0 1] [0 2] [0 2] [0] The order satisfies the following ordering constraints: [minus_active(x, y)] = [0] [0] >= [0] [0] = [minus(x, y)] [minus_active(0(), y)] = [0] [0] >= [0] [0] = [0()] [minus_active(s(x), s(y))] = [0] [0] >= [0] [0] = [minus_active(x, y)] [mark(0())] = [0] [0] >= [0] [0] = [0()] [mark(s(x))] = [2 0] x + [6] [0 0] [8] > [2 0] x + [3] [0 0] [4] = [s(mark(x))] [mark(minus(x, y))] = [0] [0] >= [0] [0] = [minus_active(x, y)] [mark(ge(x, y))] = [2] [0] > [1] [0] = [ge_active(x, y)] [mark(div(x, y))] = [0 0] y + [2 6] x + [0] [0 2] [0 0] [0] >= [0 0] y + [2 6] x + [0] [0 2] [0 0] [0] = [div_active(mark(x), y)] [mark(if(x, y, z))] = [2 0] y + [2 0] x + [2 0] z + [6] [0 2] [0 2] [0 2] [0] > [2 0] y + [2 0] x + [2 0] z + [5] [0 2] [0 2] [0 2] [0] = [if_active(mark(x), y, z)] [ge_active(x, y)] = [1] [0] >= [1] [0] = [ge(x, y)] [ge_active(x, 0())] = [1] [0] >= [1] [0] = [true()] [ge_active(0(), s(y))] = [1] [0] > [0] [0] = [false()] [ge_active(s(x), s(y))] = [1] [0] >= [1] [0] = [ge_active(x, y)] [div_active(x, y)] = [0 0] y + [1 3] x + [0] [0 2] [0 0] [0] >= [0 0] y + [1 3] x + [0] [0 1] [0 0] [0] = [div(x, y)] [div_active(0(), s(y))] = [0] [8] >= [0] [0] = [0()] [div_active(s(x), s(y))] = [1 0] x + [15] [0 0] [8] > [12] [8] = [if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0())] [if_active(x, y, z)] = [2 0] y + [1 0] x + [2 0] z + [5] [0 2] [0 1] [0 2] [0] > [1 0] y + [1 0] x + [1 0] z + [3] [0 1] [0 1] [0 1] [0] = [if(x, y, z)] [if_active(true(), x, y)] = [2 0] y + [2 0] x + [6] [0 2] [0 2] [0] > [2 0] x + [0] [0 2] [0] = [mark(x)] [if_active(false(), x, y)] = [2 0] y + [2 0] x + [5] [0 2] [0 2] [0] > [2 0] y + [0] [0 2] [0] = [mark(y)] We return to the main proof. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { minus_active(x, y) -> minus(x, y) , minus_active(0(), y) -> 0() , minus_active(s(x), s(y)) -> minus_active(x, y) , mark(0()) -> 0() , mark(s(x)) -> s(mark(x)) , mark(minus(x, y)) -> minus_active(x, y) , mark(ge(x, y)) -> ge_active(x, y) , mark(div(x, y)) -> div_active(mark(x), y) , mark(if(x, y, z)) -> if_active(mark(x), y, z) , ge_active(x, y) -> ge(x, y) , ge_active(x, 0()) -> true() , ge_active(0(), s(y)) -> false() , ge_active(s(x), s(y)) -> ge_active(x, y) , div_active(x, y) -> div(x, y) , div_active(0(), s(y)) -> 0() , div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0()) , if_active(x, y, z) -> if(x, y, z) , if_active(true(), x, y) -> mark(x) , if_active(false(), x, y) -> mark(y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^1))